X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/0a8d52f6e872c9df6f150a008456bab4557b7d09..e443cbfb0433e0a7e1551c9841b0173698151f96:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 9de2d15..dc3d3cb 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -109,7 +109,7 @@ Let us finish this paragraph by noticing that, in this paper,
 statistical perfection refers to the ability to pass the whole 
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.
-This battery can be found into the well-known TestU01 package.
+This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}.
 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 
@@ -118,7 +118,7 @@ In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on
 as a chaotic dynamical system. Such a post-treatment leads to a new category of
 PRNGs. We have shown that proofs of Devaney's chaos can be established for this
 family, and that the sequence obtained after this post-treatment can pass the
-NIST, DieHARD, and TestU01 batteries of tests, even if the inputted generators
+NIST~\cite{Nist10}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} batteries of tests, even if the inputted generators
 cannot.
 The proposition of this paper is to improve widely the speed of the formerly
 proposed generator, without any lack of chaos or statistical properties.
@@ -160,47 +160,57 @@ summarized and intended future work is presented.
 
 \section{Related works on GPU based PRNGs}
 \label{section:related works}
-In the litterature many authors have work on defining GPU based PRNGs. We do not
-want to be exhaustive and we just give the most significant works from our point
-of view. When authors mention the  number of random numbers generated per second
-we mention  it. We  consider that  a million numbers  per second  corresponds to
-1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
-
-In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
-which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
-operations. There is no mention of statistical tests nor proof that this PRNG is
-chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
-3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+Numerous research works on defining GPU based PRNGs have yet been proposed  in the
+literature, so that completeness is impossible.
+This is why authors of this document only give reference to the most significant attempts 
+in this domain, from their subjective point of view. 
+The  quantity of pseudorandom numbers generated per second is mentioned here 
+only when the information is given in the related work. 
+A million numbers  per second will be simply written as
+1MSample/s whereas a billion numbers per second is 1GSample/s.
+
+In \cite{Pang:2008:cec}  a PRNG based on  cellular automata is defined
+with no  requirement to an high  precision  integer   arithmetic  or to any bitwise
+operations. Authors can   generate  about
+3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now.
+However, there is neither a mention of statistical tests nor any proof of
+chaos or cryptography in this document.
 
 In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
-based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
+based on  Lagged Fibonacci or Hybrid  Taus.  They have  used these
 PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
 GPU. Performance of  the GPU versions are far better than  those obtained with a
-CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
-no mention that their PRNGs have chaos mathematical properties.
+CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. 
+However the evaluations of the proposed PRNGs are only statistical ones.
 
 
 Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
-PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
-(FPGA), GPU and massively parallel  processor. This study is interesting because
-it  shows the  performance  of the  same  PRNGs on  different architeture.   For
-example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
-efficient one because it provides the fastest number of generated random numbers
-per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
-with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
-the {\it Crush} test which is easier than the {\it Big Crush} test.
-
-Cuda  has developped  a  library for  the  generation of  random numbers  called
-Curand~\cite{curand11}.        Several       PRNGs        are       implemented:
-Xorwow~\cite{Marsaglia2003} and  some variants of Sobol. Some  tests report that
-the  fastest version provides  15GSample/s on  the new  Fermi C2050  card. Their
-PRNGs fail to succeed the whole tests of TestU01 on only one test.
+PRNGs on  different computing architectures: CPU,  field-programmable gate array
+(FPGA), massively parallel  processors, and GPU. This study is of interest, because
+the  performance  of the  same  PRNGs on  different architectures are compared. 
+FPGA appears as  the  fastest  and the most
+efficient architecture, providing the fastest number of generated pseudorandom numbers
+per joule. 
+However, we can notice that authors can ``only'' generate between 11 and 16GSamples/s
+with a GTX 280  GPU, which should be compared with
+the results presented in this document.
+We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only
+able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one.
+
+Lastly, Cuda  has developed  a  library for  the  generation of  pseudorandom numbers  called
+Curand~\cite{curand11}.        Several       PRNGs        are       implemented, among
+other things 
+Xorwow~\cite{Marsaglia2003} and  some variants of Sobol. The  tests reported show that
+their  fastest version provides  15GSamples/s on  the new  Fermi C2050  card. 
+But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
 \newline
 \newline
-To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
+We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
+
 This section is devoted to basic definitions and terminologies in the fields of
 topological chaos and chaotic iterations.
 \subsection{Devaney's Chaotic Dynamical Systems}
@@ -421,11 +431,12 @@ As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathd
 \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). It is as if
 $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
-\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
 
-\section{Application to pseudorandomness}
+\section{Application to Pseudorandomness}
 \label{sec:pseudorandom}
-\subsection{A First pseudorandom Number Generator}
+
+\subsection{A First Pseudorandom Number Generator}
 
 We have proposed in~\cite{bgw09:ip} a new family of generators that receives 
 two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
@@ -470,7 +481,7 @@ return $y$\;
 
 
 This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-It takes as input: a function $f$;
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques};
 an integer $b$, ensuring that the number of executed iterations is at least $b$
 and at most $2b+1$; and an initial configuration $x^0$.
 It returns the new generated configuration $x$.  Internally, it embeds two
@@ -495,7 +506,7 @@ We have proven in \cite{bcgr11:ip} that,
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 
-This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.
+This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
 
 \subsection{Improving the Speed of the Former Generator}
 
@@ -796,17 +807,26 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \section{Efficient PRNG based on Chaotic Iterations}
 \label{sec:efficient prng}
 
-In  order to  implement efficiently  a PRNG  based on  chaotic iterations  it is
-possible to improve  previous works [ref]. One solution  consists in considering
-that the  strategy used contains all the  bits for which the  negation is
-achieved out. Then in order to apply  the negation on these bits we can simply
-apply the  xor operator between  the current number  and the strategy. In
-order to obtain the strategy we also use a classical PRNG.
+Based on the proof presented in the previous section, it is now possible to 
+improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
+The first idea is to consider
+that the provided strategy is a pseudorandom Boolean vector obtained by a
+given PRNG.
+An iteration of the system is simply the bitwise exclusive or between
+the last computed state and the current strategy.
+Topological properties of disorder exhibited by chaotic 
+iterations can be inherited by the inputted generator, hoping by doing so to 
+obtain some statistical improvements while preserving speed.
 
-Here  is an  example with  16-bits numbers  showing how  the bitwise  operations
+
+Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
 are
-applied.  Suppose  that $x$ and the  strategy $S^i$ are defined  in binary mode.
-Then the following table shows the result of $x$ xor $S^i$.
+done.  
+Suppose  that $x$ and the  strategy $S^i$ are given as
+binary vectors.
+Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
+
+\begin{table}
 $$
 \begin{array}{|cc|cccccccccccccccc|}
 \hline
@@ -820,13 +840,13 @@ x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
 \hline
  \end{array}
 $$
+\caption{Example of an arbitrary round of the proposed generator}
+\label{TableExemple}
+\end{table}
 
 
 
-
-
-\lstset{language=C,caption={C code of the sequential chaotic iterations based
-PRNG},label=algo:seqCIprng}
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIprng}
 \begin{lstlisting}
 unsigned int CIprng() {
   static unsigned int x = 123123123;
@@ -847,52 +867,60 @@ unsigned int CIprng() {
 
 
 
-In listing~\ref{algo:seqCIprng}  a sequential version of  our chaotic iterations
-based PRNG is  presented.  The xor operator is  represented by \textasciicircum.
-This  function uses  three classical  64-bits PRNG:  the  \texttt{xorshift}, the
-\texttt{xor128}  and  the  \texttt{xorwow}.   In  the following,  we  call  them
-xor-like PRNGSs.   These three PRNGs are  presented in~\cite{Marsaglia2003}.  As
-each xor-like PRNG  used works with 64-bits and as our  PRNG works with 32-bits,
-the use of \texttt{(unsigned int)} selects the 32 least significant bits whereas
-\texttt{(unsigned int)(t3$>>$32)}  selects the 32 most significants  bits of the
-variable \texttt{t}.   So to produce a  random number realizes  6 xor operations
-with 6 32-bits  numbers produced by 3 64-bits PRNG.   This version successes the
-BigCrush of the TestU01 battery~\cite{LEcuyerS07}.
+In Listing~\ref{algo:seqCIprng}  a sequential version of  the proposed PRNG based on chaotic iterations
+ is  presented.  The xor operator is  represented by \textasciicircum.
+This  function uses  three classical  64-bits PRNGs, namely the  \texttt{xorshift}, the
+\texttt{xor128},  and  the  \texttt{xorwow}~\cite{Marsaglia2003}.   In  the following,  we  call  them
+``xor-like PRNGs''. 
+As
+each xor-like PRNG  uses 64-bits whereas our proposed generator works with 32-bits,
+we use the command \texttt{(unsigned int)}, that selects the 32 least significant bits of a given integer, and the code
+\texttt{(unsigned int)(t3$>>$32)}  in order to obtain the 32 most significant  bits of \texttt{t}.   
+
+So producing a  pseudorandom number needs  6 xor operations
+with 6 32-bits  numbers that are provided by 3 64-bits PRNGs.   This version successfully passes the
+stringent BigCrush battery of tests~\cite{LEcuyerS07}.
 
-\section{Efficient PRNGs based on chaotic iterations on GPU}
+\section{Efficient PRNGs based on Chaotic Iterations on GPU}
 \label{sec:efficient prng gpu}
 
-In  order to benefit  from computing  power of  GPU, a  program needs  to define
-independent blocks of threads which  can be computed simultaneously. In general,
-the larger the number of threads is,  the more local memory is used and the less
-branching  instructions are  used (if,  while, ...),  the better  performance is
-obtained  on  GPU.  So  with  algorithm  \ref{algo:seqCIprng}  presented in  the
-previous section, it is possible to  build a similar program which computes PRNG
-on   GPU.  In  the   CUDA~\cite{Nvid10}  environment,   threads  have   a  local
-identificator, called \texttt{ThreadIdx} relative to the block containing them.
-
-
-\subsection{Naive version for GPU}
-
-From the CPU version, it is possible  to obtain a quite similar version for GPU.
-The principe consists in assigning the computation of a PRNG as in sequential to
-each thread  of the  GPU.  Of course,  it is  essential that the  three xor-like
-PRNGs  used for  our computation  have different  parameters. So  we  chose them
-randomly with  another PRNG. As the  initialisation is performed by  the CPU, we
-have  chosen  to  use  the  ISAAC  PRNG~\cite{Jenkins96}  to  initalize  all  the
-parameters for  the GPU version  of our PRNG.   The implementation of  the three
-xor-like  PRNGs  is  straightforward  as  soon as  their  parameters  have  been
-allocated in  the GPU memory.  Each xor-like PRNGs  used works with  an internal
-number  $x$  which keeps  the  last  generated  random numbers.  Other  internal
-variables  are   also  used   by  the  xor-like   PRNGs.  More   precisely,  the
-implementation of the  xor128, the xorshift and the  xorwow respectively require
-4, 5 and 6 unsigned long as internal variables.
+In  order to take benefits  from the computing  power of  GPU, a  program needs  to have
+independent blocks of threads that can be computed simultaneously. In general,
+the larger the number of threads is,  the more local memory is used, and the less
+branching  instructions are  used (if,  while, ...),  the better the performances on GPU is.  
+Obviously, having these requirements in mind, it is possible to  build a program similar to 
+the one presented in Algorithm  \ref{algo:seqCIprng}, which computes pseudorandom numbers
+on   GPU.  
+To do so, we must firstly recall that in
+ the   CUDA~\cite{Nvid10}  environment,   threads  have   a  local
+identifier called \texttt{ThreadIdx}, which is relative to the block containing them.
+
+
+\subsection{Naive Version for GPU}
+
+ 
+It is possible to deduce from the CPU version a quite similar version adapted to GPU.
+The simple principle consists to make each thread of the GPU computing the CPU version of our PRNG.  
+Of course,  the  three xor-like
+PRNGs  used in these computations must have different  parameters. 
+In a given thread, these lasts are
+randomly picked from another PRNGs. 
+The  initialization stage is performed by  the CPU.
+To do it, the  ISAAC  PRNG~\cite{Jenkins96} is used to  set  all  the
+parameters embedded into each thread.   
+
+The implementation of  the three
+xor-like  PRNGs  is  straightforward  when  their  parameters  have  been
+allocated in  the GPU memory.  Each xor-like  works with  an internal
+number  $x$  that saves  the  last  generated  pseudorandom number. Additionally,  the
+implementation of the  xor128, the xorshift, and the  xorwow respectively require
+4, 5, and 6 unsigned long as internal variables.
 
 \begin{algorithm}
 
 \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
 PRNGs in global memory\;
-NumThreads: Number of threads\;}
+NumThreads: number of threads\;}
 \KwOut{NewNb: array containing random numbers in global memory}
 \If{threadIdx is concerned by the computation} {
   retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
@@ -903,37 +931,34 @@ NumThreads: Number of threads\;}
   store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }
 
-\caption{main kernel for the chaotic iterations based PRNG GPU naive version}
+\caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
 \label{algo:gpu_kernel}
 \end{algorithm}
 
-Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of  PRNG using
-GPU.  According  to the available  memory in the  GPU and the number  of threads
+Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of the proposed  PRNG on
+GPU.  Due to the available  memory in the  GPU and the number  of threads
 used simultenaously,  the number  of random numbers  that a thread  can generate
-inside   a    kernel   is   limited,   i.e.    the    variable   \texttt{n}   in
-algorithm~\ref{algo:gpu_kernel}. For example, if  $100,000$ threads are used and
-if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)}
-then   the  memory   required   to  store   internals   variables  of   xor-like
+inside   a    kernel   is   limited  (\emph{i.e.},    the    variable   \texttt{n}   in
+algorithm~\ref{algo:gpu_kernel}). For instance, if  $100,000$ threads are used and
+if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)},
+then   the  memory   required   to  store all of the  internals   variables  of both the  xor-like
 PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
-and  random  number of  our  PRNG  is  equals to  $100,000\times  ((4+5+6)\times
-2+(1+100))=1,310,000$ 32-bits numbers, i.e. about $52$Mb.
+and  the pseudorandom  numbers generated by  our  PRNG,  is  equal to  $100,000\times  ((4+5+6)\times
+2+(1+100))=1,310,000$ 32-bits numbers, that is, approximately $52$Mb.
 
-All the  tests performed  to pass the  BigCrush of TestU01  succeeded. Different
-number of threads, called \texttt{NumThreads} in our algorithm, have been tested
-upto $10$ millions.
-\newline
-\newline
-{\bf QUESTION : on laisse cette remarque, je suis mitigÃ© !!!}
+This generator is able to pass the whole BigCrush battery of tests, for all
+the versions that have been tested depending on their number of threads 
+(called \texttt{NumThreads} in our algorithm, tested until $10$ millions).
 
 \begin{remark}
-Algorithm~\ref{algo:gpu_kernel}  has  the  advantage to  manipulate  independent
-PRNGs, so this version is easily usable on a cluster of computer. The only thing
-to ensure is to use a single ISAAC PRNG. For this, a simple solution consists in
-using a master node for the initialization which computes the initial parameters
+The proposed algorithm has  the  advantage to  manipulate  independent
+PRNGs, so this version is easily adaptable on a cluster of computers too. The only thing
+to ensure is to use a single ISAAC PRNG. To achieve this requirement, a simple solution consists in
+using a master node for the initialization. This master node computes the initial parameters
 for all the differents nodes involves in the computation.
 \end{remark}
 
-\subsection{Improved version for GPU}
+\subsection{Improved Version for GPU}
 
 As GPU cards using CUDA have shared memory between threads of the same block, it
 is possible  to use this  feature in order  to simplify the  previous algorithm,
@@ -1706,7 +1731,7 @@ proving that $H$ is not secure, a contradiction.
 
 
 
-\section{A cryptographically secure prng for GPU}
+\section{A Cryptographically Secure PRNG for GPU}
 \label{sec:CSGPU}
 It is  possible to build a  cryptographically secure prng based  on the previous
 algorithm (algorithm~\ref{algo:gpu_kernel2}).   It simply consists  in replacing
